(1) to to assign river order, you can use Assign River Order tool of ArcHydroToolbox (Toolboxes\System Toolboxes\Arc Hydro Tools.tbx\Attribute Tools). This GP tool handles multiple river system at the same time, so you don't need to delete unconnected river lines.
(2) to make sure a known river line is considered in the flowdirection processing (burn-in river line), you can use the Create Drainage Line Structures tool (under Toolboxes\System Toolboxes\Arc Hydro Tools.tbx\Terrain Preprocessing\Create Drainage Line Structures)